Maxwell, Dirac and Seiberg-Witten Equations

Abstract

In this Chapter we discuss three important issues. The first is how $\mathrm{i} = \sqrt{-1}$ makes its appearance in classical electrodynamics and in Dirac theory. This issue is important because if someone did not really know the real meaning uncovered by $\mathrm{i} = \sqrt{-1}$ in these theories he may infers nonsequitur results. After that we present some ‘Dirac like’ representations of Maxwell equations. Within the Clifford bundle it becomes obvious why there are so many ‘Dirac like’ representations of Maxwell equations. The third issue discussed in this chapter are the mathematical Maxwell-Dirac equivalences of the first and second kinds and the relation of these mathematical equivalences with Seiberg-Witten equations in Minkowski spacetime $(M,\boldsymbol{\eta },D,\tau _{\boldsymbol{\eta }},\uparrow )$ which is the arena where we suppose physical phenomena to take place in this chapter. We denote by {x ^μ} coordinates in Einstein-Lorentz-Poincaré gauge associated to an inertial reference frame $\boldsymbol{e}_{0} \in \sec TM$. Moreover $\{\boldsymbol{e}_{\mu } = \frac{\partial } {\partial x^{\mu }}\} \in \sec TM,(\mu = 0,1,2,3)$ is an orthonormal basis, with $\boldsymbol{\eta }(\boldsymbol{e}_{\mu },\boldsymbol{e}_{\nu }) =\eta _{\mu \nu } =\mathrm{ diag}(1,-1,-1,-1)$ and $\{\gamma ^{\nu } = dx^{\nu }\} \in \sec \bigwedge ^{1}T^{{\ast}}M\hookrightarrow \sec \mathcal{C}\ell(M,\eta )$ is the dual basis of $\{\boldsymbol{e}_{\mu }\}$.